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Tan 20 + tan 40 + √3 × tan20 × tan40
A) √3/2
B)√3/4
C)1
D)None​

User NotAUser
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1 Answer

2 votes

Answer:

D) None

Explanation:

To solve the given expression, let's start by using the trigonometric identity:

tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)

We can rewrite the expression as follows:

tan 20 + tan 40 + √3 × tan 20 × tan 40

Using the identity, we have:

tan(20 + 40) = (tan 20 + tan 40) / (1 - tan 20 * tan 40)

tan 60 = √3

Now let's substitute tan(60) with √3:

(√3) / (1 - tan 20 * tan 40) + √3 × tan 20 × tan 40

We need to find the value of tan 20 * tan 40. To do that, let's use another trigonometric identity:

tan(A - B) = (tan A - tan B) / (1 + tan A * tan B)

tan(40 - 20) = (tan 40 - tan 20) / (1 + tan 40 * tan 20)

tan 20 = (tan 40 - tan 20) / (1 + tan 40 * tan 20)

Rearranging the equation, we get:

tan 20 + tan 20 * tan 40 = tan 40

Now, let's substitute tan 40 with (tan 20 + tan 20 * tan 40):

(√3) / (1 - tan 20 * tan 40) + √3 × tan 20 × tan 40

= (√3) / (1 - (tan 20 + tan 20 * tan 40)) + √3 × (tan 20 + tan 20 * tan 40) × tan 20 × tan 40

= (√3) / (1 - (tan 20 + tan 20 * (tan 20 + tan 20 * tan 40))) + √3 × (tan 20 + tan 20 * (tan 20 + tan 20 * tan 40)) × tan 20 × tan 40

= (√3) / (1 - (tan 20 + tan 20 * (tan 20 + tan 20 * (tan 20 + tan 20 * tan 40)))) + √3 × (tan 20 + tan 20 * (tan 20 + tan 20 * (tan 20 + tan 20 * tan 40))) × tan 20 × tan 40

By substituting recursively, we can continue this process indefinitely. Therefore, the expression doesn't have a finite solution. The answer is D) None.

User Phil Mok
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